A coefficient of an asymptotic expansion of logarithms of determinants for classical elliptic pseudodifferential operators with parameters

For classical elliptic pseudodifferential operators A(lambda) of order m > 0 with parameter lambda of weight chi > 0, it is known that log det(theta) A(lambda) admits an asymptotic expansion as lambda --> +infinity In this paper we show, with some assumptions, that the coefficient of lambda(-1/chi) can be expressed by the value of a zeta function at 0 for some elliptic psi DO on M x S-1 multiplied by m/2.